# Has anyone ever actually seen this Daniel Biss paper?

A student asked me about a paper by Daniel Biss (MIT Ph.D. and Illinois state senator) proving that "circles are really just bloated triangles." The only published source I could find was the young adult novel An Abundance of Katherines by John Green, which includes the following sentence:

Daniel [Biss] is world famous in the math world, partly because of a paper he published a few years ago that apparently proves that circles are basically fat, bloated triangles.

This is probably just Green's attempt to replicate something Biss told him about topology (an example of homotopy, perhaps).

But the statement seems to have intrigued students and non-mathematicians online. So I'm curious: has anyone seen such a paper? Is this a simplified interpretation of a real result (maybe in The homotopy type of the matroid grassmannian?).

The book with those words was published in 2006, before the retraction of Biss' major results on combinatorial differential geometry. In the cited paper, Biss had published an amazing breakthrough along the lines that Green understood, showing that certain continuous geometric objects were equivalent to discrete combinatorial objects. That is similar to, but much more general than, the relation between a geometric circle and the triangle (in the sense of graph theory, 3 vertices connected by 3 edges, oriented to go around the triangle, and not a Euclidean geometry triangle or a physical triangular plate).

I quote and annotate an excerpt from the introduction to Biss' article, to give the idea of how incredible it must have sounded at the time. Had it been correct then "world famous in the math world" would have been a good description.

[p.931]...the theory of matroid bundles is the same as the theory of vector bundles [!!]. This gives substantial evidence that a CD [Combinatorial Differential] manifold has the capacity to model many properties of smooth manifolds. To make this connection more precise, we give... a definition of morphisms that makes CD manifolds into a category admitting a functor from the category of smoothly triangulated manifolds. Furthermore, these morphisms have appropriate naturality properties for matroid bundles and hence [combinatorially defined] characteristic classes [!!!], so many maneuvers in differential topology carry over verbatim to the CD setting. This represents the first demonstration that the CD category succeeds in capturing structures contained in the smooth but absent in the topological and PL categories [!!!!!!!], and suggests that it might be possible to develop a purely combinatorial approach to smooth manifold topology [!!!].

The boldface and bracketed material are my annotations, with ! marks as a subjective rating of how amazing each statement would have been if true.

The first sentence in boldface seems to include the combinatorial construction of Pontrjagin classes, a major research problem. That would have been a big achievement, but the paper claims to do it as part of something even bigger, as the next two sentences elaborate.

The last two items in boldface, doing smooth (beyond the topological and piecewise-linear categories) topology on discrete combinatorial objects, was considered science fiction. It was not expected then or now that such a thing is possible and to do it in 20 pages must have struck many people as some kind of dream. Nevertheless, the error was apparently subtle enough to pass the reviewers, though experts were stating their skepticism about the paper soon after it was published.

The official retraction happened several years later and the author of a book published in 2006 would not necessarily have known of the problems with the paper. People were challenging the correctness of the article by 2005, and the "the problem was already acknowledged and discussed in private correspondence between experts in April 2006" but during the time the book (or his contribution to it) was written, Biss may have believed that his results were correct or could be fixed.

Biss became a politician in Illinois, and his election opponents mentioned the situation with his papers during one of the election campaigns.

• @ zyx Great explanation, just what I was looking for. Thanks! – nardol5 Oct 6 '16 at 17:33

I think the key word here is "apparently", and the full quote from Green includes a confession that he doesn't really know what he's talking about:

"I talk about [math] a lot, and I think about it a lot, but I can’t actually, like, do it."

Maybe Biss showed him a proof that circles and triangles are homeomorphic which blew Green's mind, and maybe he conflated this fairly simple result with something real that Biss had recently published. Whatever it is, it's likely the exaggeration of a clueless friend, a friend who also described Biss as "one of the best young mathematicians in America".